Our first proposed regularizer is motivated by the robustness regularizers used in the literature of certified robustness [1]}, [2]}. The regualrizer, denoted by \(\mathcal {L}_f^l\) , aims to minimize the average loss in fairness across all training data. More formally, given a training point \((x,y)\)  and NN parameters \(\theta \) , let \(\mathcal {L}(f_\theta (x), y; \theta ) = -[y \log (f_\theta (x)) + (1-y) \log (1-f_\theta (x))]\)  be the standard binary cross-entropy loss. The fairness regularizer \(\mathcal {L}_f^l\)  can then be defined as:
\(\mathcal {L}_f^l(\theta ) &=\mathbb {E}_{(x,y) \in (X,Y)} \left[ \max _{ \begin{array}{c}x^{\prime }\in D_\phi \\ d(x,x^{\prime }) = 1 \end{array} } \mathcal {L}(f_{\theta }(x^{\prime }),y;\theta ) \right]\) 
